3 SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA 325 9. With the hypotheses of Problem 8, a representative measure of a character x e B is a positive measure /x on X such that x(/) = fdp for ali/e B; such a measure p. must have total mass 1 . A quasi-representative measure of x is a complex measure B on X such that I fdd = cx(/) for all /e B, where c is a constant, or equivalently such that | fd6 = 0 for all functions /e Ker(x). If jit is a quasi-representative measure for x» then z; • //, is a quasi-representa- tive measure, for any v E B. A Jensen measure for x is defined to be a positive measure /x on X such that !og !%(/)! ^ log I/I 4*- for all/e B. (Since log |/| is /x-measurable and bounded above on X, the above integral is either finite (if log |/| is ft-integrable) or equal to — oo ; we make the convention that log lx(/)l = — oo if x(/) — 0.) (a) For each /e <dfR(X), put M(/) = sup(log |x(w)|), taken over the set of such that \u\ <>ef. Then we have M(a) = a if a e R is a constant; and M(/) + M(g) <[ M(/+ g). Also, for each inver- tible function u in B, we have M(log|w|) = log|x(w)| , and M(^w) = ^x(w) for a11 « e B- For each /e#R(X), the limit Q(/) = lim t~*M(tf) exists and is finite (Section t-* + oo 12.7, Problem 10) and has the following properties: inf/(x) g Q(/) g sup/(x); xeX xeX Q(r/) = tQ(f) for all / ^ 0; Q(/+ ^) ^ Q(/) H- Q(^). For p to be a Jensen meas- ure for x, it is necessary and sufficient that Q(/) <£ / J/x for all /e *^R(X). Hence deduce the existence of at least one Jensen measure for x (Section 13.3, Problem 2). (b) Every Jensen measure for the character x is a representative measure for x* (Observe that if /e B is invertible in B, then log lx(/)l = log I/I d^ and replace / by ef or eif.) The set 9R(x) of representative measures for x, and the set $(x) of Jensen measures for x are convex, vaguely compact subsets of MR(X). (c) If the real parts of the functions belonging to B are dense in #R(X), then there is only one representative measure for x (which is therefore the unique Jensen measure for x)- In this case B is said to be a Dirichlet algebra. (d) The algebra B denned in (15.3.9) is a Dirichlet algebra (use (7.4.2)). For the character />->/(0), the normalized Haar measure on U is the unique representative measure. 10. With the hypotheses and notation of Problem 8, put D>)= inf U6B, X(») where ft is any positive measure on X, and 0 < p < +00. (a) If 0 < p < r < + oo, thenhow that in these conditions the mapping 99 is surjective. (Let m be a maximal